170 KANSAS UNIVERSITY QUARTERLY. ” 
(1) a definite arc given to find a length of straight line equal in 
length to it, or (2) a given length of line to find an arc of assumed 
radius equal in léngth to the given line. 
The constructions for these cases depend upon the fact that any 
arc drawn about the pole of the spiral as a center and limited by 
the initial line and the spiral itself is equal to every other arc so 
drawn., That's, in {Fig; 9, arc CM *== are PH arcs eae 
this constant length of arc is known, being the constant c of the 
instrument multiplied by ao In the case of an instrument de- 
180 
signed for 6 to be measured in radians, instead of in degrees, it is 
equal the constant term of the equation. 
On each of the Universal Curves will be found a fine mark along 
the edge of the instrument which is the initial line, shown at a, 
Fig. 5. This mark is accurately laid off a distance from the pole 
of the instrument equal to the constant length of arc of the partic- 
: : 100 : ; 
ular size of curve. In the case of the size r=—_, ‘this’ length 1s 
6 
100X—2-=1 or very closely 13” 
Oss en -745) WW, Ngrirae 
(1). To rectify a given arc. Let: AB, Fig. 12, be the given are 
of which the center is O. Set the instrument as shown, marking a 
Big’ 12: 
and W. Join W to a, and draw AM from A parallel to Wa, de- 
termining M. Then OM equals the length of arc AB. 
atc AB. OA ; ees OM 
SFE OAC which by similar triangles =——. But Oa 
For Oa 
= the arc WD by the property of the instrument. 
OM= arc AB. 
(2) Given a length of line to find an are of given radius equal 
in length to the line. 
